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I.S.I. B.Stat and B.Math Entrance 2017

Let the sequence \( \{ a_n\} _{n \ge 1 } \) be defined by $$ a_n = \tan n \theta $$ where \( \tan \theta = 2 \). Show that for all n \( a_n \) is a rational number which can be written with an odd denominator.
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Consider a circle of radius 6 as given in the diagram below. Let B, C, D and E be points on the circle such that BD and CE, when extended, intersect at A. If AD and AE have length 5 and 4 respectively, and DBC is a right angle, then show that the length of BC is $$ \frac {12 + 9 \sqrt {15} }{5} $$
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Let S be the square formed by the four vertices (1, 1), (1, -1), (-1, 1), and (-1, -1). Let the region R be the set of points inside S which are closer to the center than to any of the four sides. Find the area to the region R.
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Let \( g: \mathbb{N} \to \mathbb{N} \) with g(n) being the product of the digits of n.

Let \( A = \{ 1, 2, … , n \} \). For a permutation P = { P(1) , P(2) , … , P(n) } of the elements of A, let P(1) denote the first element of P. Find the number of all such permutations P so that for that all \( i, j \in A \)